MinT - Best Known (159, 159+100, s)-Nets in Base 4

Best Known (159, 159+100, s)-Nets in Base 4

(159, 159+100, 160)-Net over F₄ — Constructive and digital

Digital (159, 259, 160)-net over F₄, using

t-expansion [i] based on digital (157, 259, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 84, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 175, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(159, 159+100, 413)-Net over F₄ — Digital

Digital (159, 259, 413)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+100, 8494)-Net in Base 4 — Upper bound on s

There is no (159, 259, 8495)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 862858 111612 727574 589706 714149 490607 239556 345521 709861 820185 101367 701715 565543 612299 014602 094954 284994 236616 668335 992258 716524 004386 210172 946099 232235 689616 > 4²⁵⁹ [i]